# Global wellposedness of the equivariant Chern–Simons–Schrödinger equation

### Baoping Liu

Peking University, Beijing, China### Paul Smith

University of California Berkeley, USA

## Abstract

In this article we consider the initial value problem for the $m$-equivariant Chern–Simons–Schrödinger model in two spatial dimensions with coupling parameter $g∈R$. This is a covariant NLS type problem that is $L_{2}$-critical. We prove that at the critical regularity, for any equivariance index $m∈Z$, the initial value problem in the defocusing case ($g<1$) is globally wellposed and the solution scatters. The problem is focusing when $g≥1$, and in this case we prove that for equivariance indices $m∈Z$, $m≥0$, there exist constants $c=c_{m,g}$ such that, at the critical regularity, the initial value problem is globally wellposed and the solution scatters when the initial data $ϕ_{0}∈L_{2}$ is $m$-equivariant and satisfies $∥ϕ_{0}∥_{L_{2}}<c_{m,g}$. We also show that $c_{m,g} $ is equal to the minimum $L_{2}$ norm of a nontrivial $m$-equivariant standing wave solution. In the self-dual $g=1$ case, we have the exact numerical values $c_{m,1}=8π(m+1)$.

## Cite this article

Baoping Liu, Paul Smith, Global wellposedness of the equivariant Chern–Simons–Schrödinger equation. Rev. Mat. Iberoam. 32 (2016), no. 3, pp. 751–794

DOI 10.4171/RMI/898